| |
Bohm's
Hidden-Variable Interpretation of Quantum Theory
Consider next a brief overview of the
hidden-variable interpretation, Bohm's means for
implementing his three-point agenda for future
microphysics.
Bohm’s hidden-variable interpretation is
the Schrödinger wave equation plus trajectories for
individual particles, as in Newton’s second law of
motion, thus rendering both the wave and particle as
real and completely causal. Measurement does not
realize the particle, and there is no wave collapse
to a particle.
Bohm postulates that there exists a
subquantum-mechanical order of magnitude containing
hidden phenomena, and that the statistical character
of the current quantum theory originates in random
fluctuations of new kinds of entities existing at
this lower subquantum-mechanical level.
Thus Heisenberg's indeterminacy principle and
his particular statistical treatment of it pertain
only to phenomena at the quantum-mechanical level.
Bohm believes that indeterminacy is a
measurement problem like the measurement problems
found in Newtonian mechanics, and that by broadening
the context of physical theory to include a
subquantum-mechanical level, it will become possible
to diminish indeterminacy below the limits set by
Heisenberg's indeterminacy principle.
Bohm states that subquantum processes may be
detectable in the domain of very high energies and
very small distances, even though at lesser energies
and greater distances the high degree of
approximation permitted by the laws of the quantum
level means that the entities at the subquantum
level cannot be playing a very significant role in
quantum-level events.
He postulates that associated with each
electron there is a particle that has a precisely
definable and continuously varying values of
position and momentum, and that is so small that at
the quantum-mechanical level it can be approximated
as a mathematical point, just as in the earliest
forms of atomic theory the atom was so described.
He also postulates that associated with the
particle there is a quantum-level wave that
oscillates in a real subquantum field, and which
satisfies the Schrödinger wave function.
In his later works he also refers to the
subquantum field as the quantum field.
In summary Bohm says he regards the
quantum-mechanical system as a synthesis of a
precisely definable particle and a precisely
definable subquantum field which exerts a force or
potential on the particle.
Bohm uses figures of speech, which he
imprecisely calls analogies, and these analogies are
not merely illustrative of fully formed thoughts,
but have had a self-consciously formative role in
his thinking.
In his Causality and Chance Bohm uses an analogy with Brownian movements of
particles in a gravitational field, and illustrates
what Heisenberg’s indeterminacy principle would
mean in terms of a subquantum-mechanical field.
In the case of Brownian motion a smoke
particle is subject to random fluctuations
originating in collisions with the atoms that exist
at a lower order of magnitude than the smoke
particle.
As a result of these random collisions the
motion of the smoke particle cannot be completely
determined by the position and velocity of the
particle at the level of the Brownian motion itself.
Bohm cites a 1933 paper by the German
physicist, R. Furth, who showed that the lack of
determination in Brownian motion is not only
qualitatively analogous to that obtained in the
quantum theory, but is also quantitatively analogous
to the mathematical form of the indeterminacy
relations.
Thus, for a short-time interval with random
fluctuations of a given magnitude in the mean
position and a given magnitude in the mean momentum,
the magnitudes satisfy a relationship involving a
constant that depends on the state of the gas, and
the relationship is mathematically analogous to
Heisenberg's indeterminacy relation involving
Planck's constant. The quantum-level force produces
a tendency to pull the particle into regions where
the subquantum-level field has its strongest
intensity, as described by Born's probability
distribution.
But this tendency is also resisted by random
motions analogous to Brownian motions, which
originate at the subquantum level.
The origin of these motions is not important;
it is sufficient they have the property such that
the average of their motions satisfies the Schrödinger
wave equation, and that they are communicated to the
particle.
The net effect of the quantum-level force and
the subquantum-level random motions in the
subquantum field is a mean distribution in a
statistical ensemble of particles, which favors the
regions where the quantum-level force field is most
intense, but which still leaves some chance for a
typical particle to spend some time in the regions
where the field is relatively weak.
This result is analogous to the classical
Brownian motion of a particle in a gravitational
field, where the random motion which tends to carry
the particle into all parts of the container, is
opposed by the gravitational field, which tends to
pull it towards the bottom of the container.
Using these concepts Bohm proposes his
alternative explanation of the two-slit experiment.
When the particle passes through a slit, it
follows an irregular path, because subquantum random
motions affect it.
After a large number of particles have passed
through the slit system with both slits open, a
pattern forms with particles accumulating on a
screen where the subquantum field intensity is
greatest due to the effects of the quantum force, as
described by Born's probability distribution.
The pattern is different if only one slit is
open, than if both are open.
Closing one of the two slits influences the
particles that pass through the open slit, because
it influences the quantum-level force felt by the
particle as it moves between the slit system and the
screen.
Thus the hidden-variable interpretation can
explain how the appearance of the wave-particle
duality originates, while the Copenhagen
interpretation requires acceptance without further
discussion of the fact that electrons enter the slit
system and appears at the screen with an
interference pattern.
In Causality
and Chance Bohm also comments on Heisenberg's
gamma-ray microscope thought experiment.
He maintains that Heisenberg's indeterminacy
principle should not be regarded as expressing the
impossibility of making measurements of unlimited
precision.
Rather it should be regarded as expressing
the incomplete degree of self-determination
characteristic only of entities that can be defined
in the quantum-mechanical level. The subquantum-mechanical
processes involving very small intervals of time and
space will not be subject to the same limitations as
those of the quantum-mechanical processes, and the
unpredictable and uncontrollable disturbances caused
by a measurement apparatus at the quantum level can
either be eliminated or be controlled and corrected.
Thus when the physicist measures processes at
the quantum-mechanical level, the process of
measurement will have the same limits on its degree
of self-determination as every other process at this
level.
But if the microphysical theory is
generalized to include the subquantum order of
magnitude, then the problem of measurement
attributed to the uncertainty principle should be
regarded not as an inherent limitation on the
precision with which it is possible to conceive the
simultaneous definition of position and momentum,
but rather as merely a practical limitation, because
measurement precision in violation of the
uncertainty relations is conceivable.
Bohm also gives another analogy with Brownian
motion.
Bohm compares quantum phenomena with Brownian
motion by describing the wave and particle as
entities that interact in a way that is essential to
their modes of being.
He says that this seems plausible, because
that fact that wave and particle are never found
separately suggests that they are both different
aspects of the some fundamentally new kind of
entity, which is likely to be quite different from a
simple wave or particle.
Thus if Brownian motion were viewed not as a
motion of particles, but as a motion of a very fine
droplet of mist, then the indeterminacy of the
droplets in a vapor at its critical temperature,
where the distinction between liquid and gaseous
states disappears, is a fluctuation in which the
droplets are always forming and disappearing.
This is an indeterminacy in the very
existence of the droplets.
Similarly at the quantum level it may be
found that the very mode of existence of the
electron is indeterminate.
The fact that the electron shows its
characteristic wave-particle duality in its behavior
suggests that the particle showing this critical
opalescence is the relevant concept of particle.
It is unclear whether or not Bohm is
attempting at this stage of his thinking in terms of
hidden variables to use this alternative Brownian
analogy to reconcile his original potentiality idea
with his newer hidden-variables idea.
In his later view in Undivided Universe potentiality is the presence of the information
in the quantum wave, which is inactive except when
the particle uses it as a guidance condition for its
movement.
Bohm says that because the subquantum level
is inadmissible in the Copenhagen interpretation,
one is restricted to making blind mathematical
manipulations with the hope that somehow one of
these manipulations will lead to a new and correct
theory.
He says that if the subquantum level is
admitted, where there are processes of very high
energy and very high frequency faster than the
processes taking place at the quantum-mechanical
level, then the details of the lower level would
become significant, and the current formulation of
the quantum theory would break down.
The creation of a particle such as a meson
may thus be conceived as a well defined subquantum-level
process, in which the field energy is concentrated
in a certain region of space in discrete amounts,
while the destruction of the particle is just the
opposite process.
At the quantum-mechanical level the precise
details of this process are not significant, and
therefore can be ignored.
This in fact is done in the current quantum
theory, which discusses the creation and destruction
of particles as merely a kind of popping in and out
of existence with special creation and destruction
operators in the mathematics.
However, with very fast high-energy processes
the results may well depend on these subquantum-mechanical
details.
And if this should be the case, then the
current quantum theory would not be adequate for the
treatment of such processes.
The original analogy used by Bohm for
developing the idea of the subquantum field is a
postulated similarity with the electromagnetic
field.
The analogy appears in his 1952 articles in Physical
Review, and reappears often in later works.
The subquantum field exerts a force on the
particle in a way that is analogous to the way that
the electromagnetic field exerts a force on a
charge.
And just as the electromagnetic field obeys
Maxwell’s equations, so too the subquantum field
obeys Schrödinger’s equation.
In both cases a complete specification of the
field at a given instant over every point in space
determines the values of the fields for all times.
And in both cases once the physicist knows
the field fluxions, he can calculate the force on a
particle, so that if he also knows the initial
position and momentum of the particle, he can
calculate its entire trajectory. Physicists are not
yet able to make experiments that localize the
position and momentum to a region smaller than that
in which the intensity of the hidden subquantum
field is applicable.
Therefore Bohm notes that they cannot yet
find clear-cut experimental evidence that the
hypothesis of the hidden variables is necessary.
There are also noted dissimilarities from the
electromagnetic wave (or negative analogies as Hesse
would say).
These dissimilarities are the distinctive
aspects of the quantum world in contrast to the
classical world. One
noteworthy dissimilarity is that the Schrödinger
equation is homogeneous while Maxwell’s equations
are inhomogeneous, with the result that unlike the
electromagnetic field, the subquantum field is not
radiated or absorbed, but simply changes its form
while its intensity remains constant.
In his later works Bohm says that the quantum
wave does not impart energy to the particle, but
instead functions as a guidance condition, while the
particle moves with its own energy.
This feature gives rise to Bohm’s concept
of active information, which he introduces in his
later book, Undivided Universe.
He describes the concept of active
information by an analogy with a radio wave which
guides a ship propelled by its own much greater
energy while piloted under the guidance of the radio
signal.
Similarly the elementary particle moves by
its own energy under the guidance of the quantum
wave.
The quantum wave does not push or pull the
particle, but rather guides it like the radio wave
guides the ship.
Bohm explains the two-slit interference
experiment in terms of active information.
If both slits are open, the quantum wave
passes through both slits while the particle passes
through only one slit, and the quantum wave contains
information about the slits.
As the particle reaches certain points in
front of the slits, it is informed to accelerate or
decelerate accordingly. Bohm says that the electron
particle with its own energy source may have a
complex and subtle inner structure, perhaps
comparable to a radio receiver.
Quite notably Bohm says the fact that the
action of the quantum potential upon the particle
depends only on its form and not on its magnitude,
implies the possibility of a strong nonlocal
connection of distant particles and a strong
dependence of the particle on its general
environmental context.
The forces between particles depend on the
wave function of the whole system, so that there is
what Bohm calls indivisible wholeness, reminiscent
of the organic wholeness of a living being in which
the very nature of each part depends on the whole.
This absence of the mutual externality and
separability of all elements which is characteristic
of the classical world, makes the quantum world very
elusive to the grasp of the physicists’
instruments.
But Bohm says it is real and more basic than
the classical world.
According to Bohm’s theory the classical
world’s autonomy emerges wherever the quantum
potential is so relatively small that it can be
neglected.
But the classical subworld is actually an
abstraction from the subtle quantum world, which is
the ultimate ground for existence.
These considerations lead to Bohm’s thesis
of the implicate order, the order in the quantum
world, which supersedes the Cartesian order of the
classical world and its mathematics.
In several of his publications Bohm uses the
analogy of the lens and the hologram to illustrate
the implicate order in ordinary experience.
The classical world is like a lens, which
produces an approximate correspondence of points on
an object to points on an image.
In contrast the quantum world is like a
hologram, in which each region of the hologram makes
possible an image of the whole object.
The hologram does not look like the
represented object at all, but rather the image is
implicit or enfolded.
Bohm adds that the term enfolded is not
merely a metaphor, but is to be taken literally, and
that the order in the hologram is implicate. He also
says that there are algebras of the implicate order,
and he exemplifies some in his Undivided
Universe.
Bohm’s most noteworthy analogy is given in
the third chapter of Undivided
Universe, where he develops the basic principles
of his ontological interpretation in the context of
the one-body system.
He begins with what is known as the WKB
approximation for the classical limit in quantum
mechanics, and concludes with an equation of motion
containing separate terms for both classical and
quantum forces, and describing the electron as a
particle that has a well defined position that
varies continuously, is causally determined, and is
never separated from a new type of quantum field
that affects the particle.
Peat, Bohm’s editor, explains Bohm’s
development of this analogy in his biography’s
seventh chapter titled “Hidden Variables”.
Bohm later rejected Einstein’s idea that
the probabilistic results of quantum theory are the
result of underlying deterministic motions of
smaller particles, as in the Brownian motion
analogy.
Bohm knew that something analogous to the
quantum theory’s wave-particle ambiguity already
existed in classical physics.
In the nineteenth century the Irish
mathematician W.R. Hamilton had shown that it is
mathematically possible to recast Newton’s laws
about the movement of particles into a description
involving waves.
Bohm also knew that Hamilton’s approach is
used in quantum theory as an approximation which
simplifies calculations, the WKB approximation,
which Peat says has a position midway between
classical and quantum mechanics with its assumption
that quantum particles move along actual
trajectories.
Unlike most physicists, Bohm took the WKB
approximation realistically instead of
instrumentally, i.e. as merely a convenient
approximation.
Peat reports that Bohm’s strategy was to
ask what would have to be added to Hamilton’s
approach in order to transform this mathematical
approximation technique into an equation that can
reproduce all the results of quantum theory exactly,
and that Bohm’s answer was to introduce his
radically new quantum potential, in order to explain
all the nonclassical effects.
Peat reports that Bohm thus dispensed with
metaphysical ideas like Heisenbergian potentialities
and actualities, collapsing wave functions, and
irreducible probabilities.
Bohm’s
Critique of Heisenberg’s Copenhagen Interpretation
Shortly after the publication of
Heisenberg’s
Physics and Philosophy: The Revolution in Modern
Science (1958) Bohm wrote an article in
The British Journal for the Philosophy of Science
(February, 1962) titled “Classical and
Nonclassical Concepts in the Quantum Theory.”
In a footnote Bohm comments that his article
was originally planned as a review of
Physics
and Philosophy, but since he and Heisenberg had
on previous occasions criticized one another’s
views, Bohm decided to subtitle his article “An
Answer to Heisenberg’s
Physics and Philosophy.”
On the first page of his article Bohm says
that since Heisenberg’s book presents the basic
features of the Copenhagen interpretation in such a
clear light that it constitutes a useful basis on
which further criticisms can be developed.
And in this paper Bohm sets forth his own
criticisms, one ontological and the other semantical.
In summary Bohm’s ontological criticism is
that in the exposition of his Copenhagen
interpretation Heisenberg introduces ideas that are
subjectivist and inconsistent.
Bohm believes that in expounding his doctrine
of potentia
Heisenberg states that whereas possibilities can
exist outside the human mind, physical actuality can
only exist when someone perceives it.
To assess Bohm’s criticism, it is necessary
firstly to examine Heisenberg’s own statements
about the role of subjectivism in quantum theory.
Heisenberg’s version of the Copenhagen
interpretation is set forth in the third chapter of
his Physics
and Philosophy, which is titled “The
Copenhagen Interpretation of Quantum Theory.”
He begins by comparing experiments in
classical and quantum physics.
In both types of experiments there are errors
of measurement observation, which can be described
by probability functions.
The error is not a property of the observed
system, but is the experimenter’s ignorance or
lack of knowledge of the true measurement.
Thus Heisenberg invokes a subjective
interpretation of the probability function
describing measurement error.
He then states several times in his
exposition that in the case of a quantum experiment
the probability function combines both objective and
subjective elements in the experimental situation,
or as he also says, it represents both statements of
a fact and statements of our knowledge of the fact.
The statement of our incomplete knowledge of
the fact is the measurement error, which is
subjective, and it may be different for different
experimenters, presumably because the different
experimenters do not make exactly the same errors
when making their measurements.
And he comments that the subjective element
in the probability function may be practically
negligible as compared with the objective element,
and the physicist can then speak of a pure case.
The statement of fact is a statement about
possibilities or tendencies, and he references
Aristotle’s concept of potentia.
The potentia
or potential is completely objective and does not
depend on any observer.
The realization of the potential, the
transition from the possible to the actual, takes
place during the act of observation, as soon as the
of object interacts with the measuring device.
Heisenberg explicitly issues some caveats,
namely that this transition applies to the physical
and not to the psychical act of observation, that it
is not connected with the act of registration of the
result by the mind of the observer, and that quantum
theory does not contain genuinely subjective
features, because it does not introduce the mind of
the physicist as a part of the atomic event.
These comments would suggest that Heisenberg
wishes to preclude any metaphysical idealism such as
Berkeley’s esse est percipi.
But Bohm argues that these caveats are
inconsistent with Heisenberg’s preceding
statements about subjectivism in these passages.
|
|
